Through The Ages-Part XVI - Military: A Numbers Game

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Military Actions and Military Cards
Players have more control over their civil actions than Military cards.  The cards may not appear at opportune times, but through manipulation of the card row, players can choose to pick cards from the card row before other players.  It may cost more civil actions than the player would prefer, but the player has the option to take the card or not.  Military and Event cards differ in that a player's options are limited to what the player draws randomly from the military deck.

A player can pursue a military domination strategy, but there is no guarantee the player will draw the military cards to support the strategy.
Understanding Odds
The possible combination of odds is nearly impossible to figure out in any usable form.  Instead, it is possible to show the different odds of drawing a card(s) under various conditions.  Important to note is that in a 4-player game the deck of event cards will most likely be reshuffled.  In Age I a reshuffle is almost guaranteed.  The turn of the reshuffle will differ depending on the number of military actions the players manage to develop, but it is important to realize any cards which are discarded will become available later in the turn.

A second item to take note of involves when Military cards are viable.  Military cards are not drawn until Turn 2.  Furthermore, any military cards drawn on the last turn of the game (Turn 20 using our example) will not be played, with the possible exception of Defense cards.
The Baseline
The following diagram provides a baseline of drawing a single specific card each turn if no player modifies their original 2 Military actions.  Thus, on the third turn of a game there is only a 4.26% chance of drawing Barbarians by the first player. Using only Age I as our example, we can see a general trend upwards in odds as time goes on.  Overall, this gradually increases for each player, until turn 6 when the odds reach 10%.  After this point, the increase rises dramatically.
We can also see how each player later in the turn has improved odds of drawing the specific copy of the card over the player earlier in the turn order. It doesn't seem greatly significant until turn 7. By turn 8 it is guaranteed that if a card was not drawn, it will be drawn by Player4 as there will be only one card left in the deck to draw. Additionally, in this scenario, Player 4 will reshuffle the discarded cards and draw one of them into their hand.

There are two things which we will deviate for comparison purposes.  First, we will deviate the odds if one player draws an additional military card starting on turn 2.  This is possible with Julius Caesar, a popular leader.  Second we will modify the number of copies of a card available in the deck: such as Medieval Army.
Increasing Military Actions
This next chart involves modifying the military actions of the first player while holding the other three players at 2 actions each, the first player sees a noticeable increase in their odds.
Three things become very clear.  First, the "reshuffle point" moves forward as the number of cards drawn increases.  This impacts decision making only if a player discards a card which they later "want back".
Second, the increase in odds is slightly less than doubled, until the last turn.  From our examination of the baseline, these odds would only be greater for those later in the turn if they increase their military actions. Similarly, the military actions decrease the odds of the other players, but only by a
Increasing Copies of Cards
The second variable to review is the increasing the copies of cards in the Military Deck.  Through the Ages Military cards have four levels of occurrences.  Most cards are singles, like Events and Treaties, with only one copy of the card in the deck.  Aggressions have two to four copies of the cards, based on the Age.  Tactics cards have from 1 to 4 copies, depending on the Age and the card.  Lastly, there are 6 copies of the Defense/Colonization cards in each Age.
From this data, the following graph was formed to represent the odds of drawing at least one (or more) of the cards each turn depending if each player only has 2 Military Actions.
Conclusions

Wars don't appear
until Age II, and even then
they are Rare until Age III

Military cards really are dependent on the "luck of the draw".  Players can still manipulate that luck, to some degree.  First, increasing Military actions greatly improves the odds of drawing a desired military card, although the odds remain relatively low until later turns.  This as the effect of decreasing the time required for a "reshuffle" of the deck.  This may not seem very important, but if a player discards an undesired card early, the card may make a reappearance later in the turn.

Secondly, depending on the number of occurrences of the card, the player's odds increase.  Overall, the increase in occurrences has a greater impact than the increase in Military actions.  Players only have control over their Military Actions.


Although interesting, I'm not sure the information is very useful.  Having more Military Actions provides options, which is one reason Julius Caesar is popular.  But there is little control over what appears and when it will appear.  However, many events depend on Military Strength, so military cannot be ignored.